Explanation Given: The parametric equation for the variable x is as follows. x = t 3 The parametric equation for the variable y is as follows. y = t 2 The value of t will range from 0 to 1 . Calculation: The surface area of the surface obtained by rotating curve about the x axis. S = ∫ 0 1 2 π y ( d x d t ) 2 + ( d y d t ) 2 d t The value t ranges from 0 to 1 . Differentiate the variable x with respect to t . d x d t = 3 t 2 Differentiate the variable y with respect to t . d y d t = 2 t Write the length of the curve formula. S = ∫ 1 3 2 π y ( d x d t ) 2 + ( d y d t ) 2 d t Substitute ( 3 t 2 ) for d x d t , ( t 2 ) for y , and ( 2 t ) for d y d t in the above equation. S = ∫ 0 1 2 π y ( d x d t ) 2 + ( d y d t ) 2 d t = ∫ 0 1 2 π ( t 2 ) ( 3 t 2 ) 2 + ( 2 t 2 ) 2 d t = ∫ 0 1 2 π t 2 9 t 2 + 4 d t S = 2 π ∫ 0 1 t 2 9 t 2 + 4 d t Convert the variable t inside the integration into variable u . The expression ( 9 t 2 + 4 ) is u and 18 t d t is d u . Convert the limits from variable t to variable u . Substitute 0 for t in equation ( 9 t 2 + 4 ) = u . ( 9 t 2 + 4 ) = u ( 9 ( 0 ) 2 + 4 ) = u u = 4 Substitute 1 for t in equation ( 9 t 2 + 4 ) = u

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Chapter 10.2, Problem 61E

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To find: The surface area of the curve for the parametric equation x=t3 and y=t2.

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Chapter 10.2, Problem 60E

Chapter 10.2, Problem 62E

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Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

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The surface area of the curve for the parametric equation x = t 3 and y = t 2 .

Solution Summary: The author calculates the surface area of the curve for the parametric equation x = t3, and y =

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To find: The surface area of the curve for the parametric equation x=t3 and y=t2.

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Chapter 10 Solutions